If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to (1) - 1080 (2) - 1020 (3) - 1200 (4) - 120
One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is (1) $\frac { 2 } { 3 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 4 } { 9 }$ (4) $\frac { 3 } { 5 }$
Let the position vectors of the vertices $A , B$ and $C$ of a tetrahedron $A B C D$ be $\hat { \mathbf { i } } + 2 \hat { \mathbf { j } } + \hat { \mathbf { k } } , \hat { \mathbf { i } } + 3 \hat { \mathbf { j } } - 2 \hat { k }$ and $2 \hat { i } + \hat { j } - \hat { k }$ respectively. The altitude from the vertex $D$ to the opposite face $A B C$ meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac { \sqrt { } \overline { 110 } } { 3 }$ and the volume of the tetrahedron is $\frac { \sqrt { 805 } } { 6 \sqrt { 2 } }$, then the position vector of $E$ is (1) $\frac { 1 } { 12 } ( 7 \hat { \mathbf { i } } + 4 \hat { \mathbf { j } } + 3 \hat { k } )$ (2) $\frac { 1 } { 2 } ( \hat { i } + 4 \hat { j } + 7 \hat { k } )$ (3) $\frac { 1 } { 6 } ( 12 \hat { i } + 12 \hat { j } + \hat { k } )$ (4) $\frac { 1 } { 6 } ( 7 \hat { \mathrm { i } } + 12 \hat { \mathrm { j } } + \hat { \mathrm { k } } )$
Marks obtains by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is (1) 52 (2) 48 (3) 44 (4) 40
Let a curve $y = f ( x )$ pass through the points $( 0,5 )$ and $\left( \log _ { e } 2 , k \right)$. If the curve satisfies the differential equation $2 ( 3 + y ) e ^ { 2 x } d x - \left( 7 + e ^ { 2 x } \right) d y = 0$, then $k$ is equal to (1) 4 (2) 32 (3) 8 (4) 16
Let $P$ be the foot of the perpendicular from the point $Q ( 10 , - 3 , - 1 )$ on the line $\frac { x - 3 } { 7 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { - 2 }$. Then the area of the right angled triangle $P Q R$, where $R$ is the point $( 3 , - 2,1 )$, is (1) $9 \sqrt { 15 }$ (2) $\sqrt { 30 }$ (3) $8 \sqrt { 15 }$ (4) $3 \sqrt { 30 }$
Let the $\operatorname { arc } A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac { \text { length of } \operatorname { arc } A B } { \text { length of } \operatorname { arc } B C } = \frac { 1 } { 5 }$, and $\overrightarrow { O C } = \alpha \overrightarrow { O A } + \beta \overrightarrow { O B }$, then $\alpha + \sqrt { 2 } ( \sqrt { 3 } - 1 ) \beta$ is equal to (1) $2 \sqrt { 3 }$ (2) $2 - \sqrt { 3 }$ (3) $5 \sqrt { 3 }$ (4) $2 + \sqrt { 3 }$
If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to (1) 6 (2) 10 (3) 20 (4) 12
The number of words, which can be formed using all the letters of the word ``DAUGHTER'', so that all the vowels never come together, is (1) 36000 (2) 37000 (3) 34000 (4) 35000
Let $\mathbf { R } = \{ ( 1,2 ) , ( 2,3 ) , ( 3,3 ) \}$ be a relation defined on the set $\{ 1,2,3,4 \}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: (1) 10 (2) 7 (3) 8 (4) 9
Let the area of a $\triangle P Q R$ with vertices $P ( 5,4 ) , Q ( - 2,4 )$ and $R ( a , b )$ be 35 square units. If its orthocenter and centroid are $O \left( 2 , \frac { 14 } { 5 } \right)$ and $C ( c , d )$ respectively, then $c + 2 d$ is equal to (1) $\frac { 8 } { 3 }$ (2) $\frac { 7 } { 3 }$ (3) 2 (4) 3
Let $\left| \frac { \bar { z } - i } { 2 \bar { z } + i } \right| = \frac { 1 } { 3 } , z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $( 0,0 ) , \mathrm { C }$ and $( \alpha , 0 )$ is 11 square units, then $\alpha ^ { 2 }$ equals: (1) 50 (2) 100 (3) $\frac { 81 } { 25 }$ (4) $\frac { 121 } { 25 }$
Let the circle $C$ touch the line $x - y + 1 = 0$, have the centre on the positive x-axis, and cut off a chord of length $\frac { 4 } { \sqrt { 13 } }$ along the line $- 3 x + 2 y = 1$. Let H be the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2 \alpha ^ { 2 } + 3 \beta ^ { 2 }$ is equal to $\_\_\_\_$
If the equation $\mathrm { a } ( \mathrm { b} - \mathrm { c } ) \mathrm { x } ^ { 2 } + \mathrm { b } ( \mathrm { c } - \mathrm { a } ) \mathrm { x } + \mathrm { c } ( \mathrm { a } - \mathrm { b } ) = 0$ has equal roots, where $\mathrm { a } + \mathrm { c } = 15$ and $\mathrm { b } = \frac { 36 } { 5 }$, then $a ^ { 2 } + c ^ { 2 }$ is equal to
If the set of all values of a, for which the equation $5 x ^ { 3 } - 15 x - a = 0$ has three distinct real roots, is the interval $( \alpha , \beta )$, then $\beta - 2 \alpha$ is equal to $\_\_\_\_$
If the area of the larger portion bounded between the curves $x ^ { 2 } + y ^ { 2 } = 25$ and $y = | x - 1 |$ is $\frac { 1 } { 4 } ( b \pi + c ) , b , c \in N$, then $b + c$ is equal to